**Abstract: **

We consider barotropic compressible Navier−Stokes equations in the interval $[0,\pi]$ with homogeneous Dirichlet boundary conditions. Our result is the following: given two sufficiently close subintervals $I$ and $J$ of $(0,1)$, we construct a smooth external force $f$ in the momentum equation supported in $(1,\pi)$ such that the flow map moves $I$ exactly onto $J$ in a given time $T>0$. The essential point in the proof is to find two external forces $f_1$ and $f_2$ that have "independent" stretching effect on $I$. Such forces are constructed using the linearized adjoint system and the independence is proved using a unique continuation property which we prove based on Fourier analytic techniques. This is a joint work with Franck Sueur (Université du Luxembourg) and Gastón Vergara-Hermosilla (Université Paris-Saclay).

**Biography: **

Kai Koike is an assistant professor at Department of Mathematics, Tokyo Institute of Technology. He earned his PhD from Keio University in 2019. He has worked on topics including long-time behavior of fluid−structure interaction problems for some kinetic and compressible fluid models.

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